# geometric series

A *geometric series ^{}* is a series of the form

$\sum _{i=1}^{n}}a{r}^{i-1$ |

(with $a$ and $r$ real or complex numbers^{}). The partial sums of a geometric series are given by

${s}_{n}={\displaystyle \sum _{i=1}^{n}}a{r}^{i-1}={\displaystyle \frac{a(1-{r}^{n})}{1-r}}.$ | (1) |

An *infinite geometric series* is a geometric series, as above, with $n\to \mathrm{\infty}$. It is denoted by

$\sum _{i=1}^{\mathrm{\infty}}}a{r}^{i-1$ |

If $|r|\ge 1$, the infinite geometric series diverges. Otherwise it converges to

$\sum _{i=1}^{\mathrm{\infty}}}a{r}^{i-1}={\displaystyle \frac{a}{1-r}$ | (2) |

Taking the limit of ${s}_{n}$ as $n\to \mathrm{\infty}$, we see that ${s}_{n}$ diverges if $|r|\ge 1$. However, if $$, ${s}_{n}$ approaches (2).

One way to prove (1) is to take

${s}_{n}=a+ar+a{r}^{2}+\mathrm{\cdots}+a{r}^{n-1}$ |

and multiply by $r$, to get

$r{s}_{n}=ar+a{r}^{2}+a{r}^{3}+\mathrm{\cdots}+a{r}^{n-1}+a{r}^{n}$ |

subtracting the two removes most of the terms:

${s}_{n}-r{s}_{n}=a-a{r}^{n}$ |

factoring and dividing gives us

${s}_{n}={\displaystyle \frac{a(1-{r}^{n})}{1-r}}$ |

$\mathrm{\square}$

Title | geometric series |
---|---|

Canonical name | GeometricSeries |

Date of creation | 2013-03-22 12:05:37 |

Last modified on | 2013-03-22 12:05:37 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 16 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 40A05 |

Related topic | GeometricSequence |

Related topic | ExampleOfAnalyticContinuation |

Related topic | ApplicationOfCauchyCriterionForConvergence |

Defines | infinite geometric series |